Large Deviations and Metastability

Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
CONTENTS......Page 9
PREFACE......Page 13
Introduction......Page 19
1.1 Cramér–Chernoff theorem on R......Page 21
1.2 Abstract formulation......Page 38
Large deviations and evaluation of integrals......Page 42
Contraction principle......Page 45
1.3 Sanov theorem for finite ‘alphabets’......Page 46
Beyond the empirical measure. Finite words......Page 49
The empirical process......Page 53
1.4 Cramér theorem on Rd......Page 57
1.5 Cramér theorem in infinite dimensional spaces......Page 62
1.6 Empirical measures. Sanov theorem......Page 74
2 Small random perturbations of dynamical systems. Basic estimates of Freidlin and Wentzell......Page 82
2.1 Brownian motion......Page 83
Some basic properties of Brownian motion......Page 89
Markov property......Page 90
A few sample path properties......Page 92
2.2 Itô’s integral......Page 95
Itô’s formula......Page 100
Girsanov theorem......Page 103
2.3 Itô’s equations......Page 105
2.4 Basic Freidlin and Wentzell estimates......Page 108
2.5 Freidlin and Wentzell basic estimates. Variable diffusion coefficients......Page 116
2.6 Exit from a domain......Page 122
3.1 Large deviations for dependent variables. Gärtner–Ellis theorem......Page 136
3.2 Large deviations for Markov chains......Page 143
Beyond the empirical measure; words of length 2......Page 146
Empirical process......Page 150
3.3 A brief introduction to equilibrium statistical mechanics......Page 152
3.3.1 Orthodicity. Ensembles......Page 155
3.3.2 Lattice systems: Gibbs measures, thermodynamic limit......Page 170
The standard Ising model: low temperature phase transition......Page 183
3.4 Large deviations for Gibbs measures......Page 194
3.4.1 Large deviation principle for the empirical field......Page 196
Large deviations for empirical averages (level 1)......Page 208
Large deviations for empirical measures (level 2)......Page 211
Boltzmann principle of equivalence of ensembles......Page 213
4.1 The van der Waals–Maxwell theory......Page 216
4.1.1 The Curie–Weiss theory......Page 223
4.1.2 Kac potentials......Page 226
4.1.3 Metastability and the Gibbsian approach......Page 228
4.1.4 Classical theory of nucleation......Page 231
4.1.5 The point of view of the evolution of the ensembles......Page 236
4.2 The pathwise approach: basic description......Page 243
4.3 The Curie–Weiss Markov chain......Page 244
4.4 The Harris contact process......Page 260
Monotonicity and attractiveness......Page 264
Harris inequality......Page 266
Dynamical phase transition......Page 268
Ergodic behaviour......Page 270
4.4.2 The metastable behaviour of the finite contact process for d = 1......Page 272
Metastability for higher dimensional contact process......Page 295
Local mean field model on the circle......Page 298
Glauber dynamics for Kac interaction......Page 299
Random field Curie–Weiss models......Page 300
5 Metastability. Models of Freidlin and Wentzell......Page 305
5.1 The coupling method of Lindvall and Rogers......Page 307
5.2 The case of a double well potential......Page 312
5.3 General case. Asymptotic magnitude of the escape time......Page 320
5.4 Cycles and asymptotic unpredictability of the escape time......Page 337
5.5 Notes and comments......Page 347
6 Reversible Markov chains in the Freidlin–Wentzell regime......Page 353
Introduction......Page 355
6.1 Definitions and notation......Page 356
6.2 Main results......Page 366
6.3 Restricted dynamics......Page 385
6.4 Conditional ergodic properties......Page 388
6.5 Escape from the boundary of a transient cycle......Page 390
6.6 Asymptotic exponentiality of the exit time......Page 393
6.7 The exit tube......Page 396
6.8 Decomposition into maximal cycles......Page 403
6.9 Renormalization......Page 406
6.10 Reduction and recurrence......Page 413
6.11 Asymptotics in probability of tunnelling times......Page 415
Introduction......Page 417
7.1 The standard stochastic Ising model in two dimensions......Page 419
7.2 The local minima......Page 424
7.3 Subcritical and supercritical rectangles......Page 428
7.4 Subcritical configurations and global saddles......Page 430
7.5 Alternative method to determine…......Page 442
7.6 Extensions and generalizations......Page 445
7.7 The anisotropic ferromagnetic Ising model in two dimensions......Page 446
7.8 Alternative approach to study of the J1J2 model......Page 451
7.9 The ferromagnetic Ising model with nearest and next nearest neighbour interactions in two dimensions......Page 456
7.10 The ferromagnetic Ising model with alternating field......Page 461
7.11 The dynamic Blume–Capel model. Competing metastable states and different mechanisms of transition......Page 465
7.12 Metastability in the two-dimensional Ising model with free boundary conditions......Page 472
7.13 Standard Ising model under Kawasaki dynamics......Page 474
7.13.2 Dynamics in three dimensions......Page 478
7.13.3 Results for local Kawasaki dynamics in two dimensions......Page 483
7.13.4 Results for local Kawasaki dynamics in three dimensions......Page 485
Results for Glauber dynamics in three dimensions......Page 487
7.14 Metastability for reversible probabilistic cellular automata......Page 494
7.15 Discussion of the results of Bovier and Manzo......Page 498
7.16 Metastability at infinite volume and very low temperature for the stochastic Ising model......Page 501
7.17 Metastability at infinite volume and very low temperature for the dynamic Blume–Capel model......Page 504
7.18 Metastability for the infinite volume stochastic Ising model at T < Tc in the limit…......Page 506
7.19 Applications......Page 508
7.20 Related fields......Page 509
REFERENCES......Page 511
Index......Page 525